🎲 Dice Expectation and Probabilities in Craps

✅ What Is Dice Expectation?

Expectation (or expected value) is a mathematical concept that tells us the average result we can expect from a random event over time. In the case of dice, it’s the average value you would get if you rolled the dice thousands or millions of times.

For One Die:

• Possible outcomes: 1, 2, 3, 4, 5, 6
• Each has an equal probability: $\frac{1}{6}$
• Expected Value (EV) = $\frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$

For Two Dice (Used in Craps):

• Possible sums: 2 through 12
• Total combinations: 36 (6 sides × 6 sides)
• Some sums are more likely than others. For example:
• 7 can be rolled in 6 ways (e.g., 1+6, 2+5, 3+4, etc.)
• 2 or 12 only in 1 way each (1+1 or 6+6)

So, the expected value for a roll of two dice is:

That’s the average result you'd see over many rolls.

🎯 Probability Breakdown (Craps Focused)

These probabilities form the foundation of craps — the rules and payouts are structured around them.

🧠 Conclusion: Do Dice Perform as Theory Predicts?

Yes — in the long run, dice perform very closely to what theory and probability predict. Here’s why:

• Randomness: Properly made and rolled dice are fair, meaning each side has an equal chance of landing.
• Law of Large Numbers: Over many rolls, the actual outcomes converge to the theoretical probabilities.
• For example, if you roll dice 36,000 times, you’ll see a 7 appear close to 6,000 times (about 16.67%).

Real-World Note:

Short-term deviations (like rolling a 7 five times in a row) are normal and expected in randomness — but over time, these balance out.

✅ Summary

• Dice expectation tells us what outcome we should expect on average.
• In craps, the most common total is 7, with an expected value of 7 for two dice.
• The probabilities of each dice combination are known and consistent.
• Over time, real dice rolls closely match theoretical expectations, confirming the reliability of mathematical models in games of chance like craps.

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